We can prove exactly the same type of result replacing \(\text{{\sf CH}}\)  by \(2^{\aleph _0}>\aleph _2\) .
Specifically Moore introduced in [1]} a \(\Pi _2\) -sentence \(\theta _{\mathrm {Moore}}\)  for \(\in _{\Delta _0}\)  to show the existenc of a definable well order of the reals in type \(\omega _2\)  in models of the bounded proper forcing axiom.
We can use \(\theta _{\mathrm {Moore}}\) 
as follows:
